Abstract
Rough set was defined by Pawlak in 1982. Concept of soft set was proposed as a mathematical tool to cope with uncertainty and vagueness by Molodtsov in 1999. Soft sets were combined with rough sets by Feng et al. in 2011. Feng et al. investigated relationships between a subset of initial universe of soft set and a soft set. Feng et al. defined the upper and lower approximations of a subset of initial universe over a soft set. In this study, we firstly define concept of soft class and soft class operations such as union, intersection, and complement. Then we give some properties of soft class operations. Based on definition and operations of soft classes, we define lower and upper approximations of a soft set. Subsequently, we introduce concept of soft rough class and investigate some properties of soft rough classes. Moreover, we give a novel decision making method based on soft class and present an example related to novel method.
1. Introduction
The concept of soft set was introduced by Molodtsov [1] in 1999 as a general mathematical tool for dealing with problems involving uncertain data. Maji et al. [2] defined some concepts and operations on soft sets such as soft subset, soft equality, soft union, soft intersection, and soft complement. Çağman and Enginoğlu [3] redefined soft set operations suggested by Maji et al. [2] and developed a decision making method called uni-int decision making method. Çağman [4] made some contributions to the theory of soft sets to fill gaps of former definition and operations.
Rough set theory was proposed by Pawlak [5] as an alternative approach to fuzzy sets theory and tolerance theory and has been applied successfully to a lot of fields such as machine learning, pattern recognition, and data mining. Dubois and Prade [6] defined lower and upper approximations of a fuzzy set to extend concept of rough set and they proposed the rough fuzzy sets. Soft sets were combined with fuzzy sets and rough sets by Feng et al. [7]. In 2011, Feng et al. [8] introduced soft rough approximation space and soft rough set based on the novel granulation structures called soft approximation spaces and presented basic properties of soft rough approximations supported by some illustrative example. They also defined some new types of soft sets such as full soft sets, intersection complement softs set, and partition soft sets. Meng et al. [9] proposed a new soft rough set model and derived its properties. They also established a more general model called soft rough fuzzy set. Irfan Ali [10] discussed concept of approximation space associated with each parameter in a soft set and defined an approximation space associated with the soft sets and established connection between soft set, fuzzy soft set, and rough sets. Feng [11] gave an application of soft rough approximations in multicriteria group decision making problems. Zhang [12] defined a new rough set model and investigated its some fundamental properties. He also presented a decision making method for intuitionistic fuzzy soft sets based on this new rough set approach. Zhang [13] studied parameter reduction of fuzzy soft sets based on soft fuzzy rough set and defined some new concepts such as lower soft fuzzy rough approximation operator and upper soft fuzzy rough approximation operator. To find approximation of a set, Shabir et al. [14] proposed modified soft rough sets. Sun and Ma [15] proposed a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set. They also defined concept of the pseudofuzzy binary relation and based on this concept they defined the soft fuzzy rough lower and upper approximation operators of any fuzzy subset in the parameter set. In this paper, we define concept of soft class and soft class operations based on decision makers set and investigate some fundamental properties of soft class operations. Then, we define soft rough class approximations and soft rough class and investigate some properties of them. Furthermore, we present a method to evaluate the decision makers and give an example to illustrate the process of this method. Proposed method can be used in many areas such as industrial engineering, economy, and social sciences. In particular, in industrial engineering, it can be used effectively for Quality Lifecycle Management and Choosing Product.
2. Preliminary
Let be an initial universe, let be the universe of all possible parameters related to the objects in , and let be power set of .
Definition 1 (see [1]). Consider a nonempty set such that . A pair is called a soft set over , where is a mapping given by .
In this paper, we will use the following definition given by Çağman [4] for basic set operations on soft sets.
Definition 2 (see [4]). A soft set over is a set valued function from to . It can be written as a set of ordered pairs:Note that if , then the element will not appear in soft set . Set of all soft sets over will be denoted by .
Example 3. Let be the universe containing eight houses and let be the set of parameters. Here, stand for the parameters “modern,” “with parking,” “expensive,” “cheap,” “large,” and “near to city,” respectively. Then, the following soft sets are described by Mr. A and Mr. B who want to buy a house, respectively:
Definition 4 (see [4]). Let . Then,(1)if , for all , is said to be a null soft set, denoted by ;(2)if , for all , is said to be absolute soft set, denoted by ;(3) is soft subset of , denoted by , if for all ;(4), if and ;(5)soft union of and , denoted by , is a soft set over and is defined by such that for all ;(6)soft intersection of and , denoted by , is a soft set over and is defined by such that for all ;(7)soft complement of is denoted by and is defined by such that for all .
Example 5. Let us consider soft sets and given in Example 3. Then,
Definition 6 (see [8]). Let be a soft set over . Then, the pair is called soft approximation space. Based on the soft approximation space , we define the two operations,assigning to every subset two sets and , which are called the soft -lower approximation and the soft -upper approximation of , respectively. In general, we refer to and as soft rough approximations of with respect to . Moreover, the setsare called the soft -positive region, the soft -negative region, and the soft -boundary region of , respectively. If , is said to be soft -definable; otherwise is called a soft -rough set.
Example 7 (see [8]). Let , let , and let . Let be a soft set over given by Table 1 and the approximation space .
For , we have and . Thus and is a soft -rough set. Note that in this case. Moreover, it is easy to see that , , and . On the other hand, one can consider . Since , by definition, is a soft -definable set.
3. Soft Classes
In this section, we define concept of soft class and soft class operations. Also we obtain some basic properties of soft class operations.
Definition 8. Let be a parameter set, let be an initial universe, and let be a set of decision makers. Indexed class of soft sets is called a soft class and is denoted by .
If, for any , , the soft set does not appear in soft class .
Throughout this study , , and denote parameter set, initial universe, and decision makers set, respectively.
From now on, all soft classes over parameter set , initial universe , and decision makers set will be denoted by .
Example 9. Let be a parameter set, let be an initial universe, and let be a set of decision makers. If we consider soft sets given as then is a soft class. We can represent a soft class in tabular form as shown in Table 2.
Definition 10. Let . If, for all , , then is called an empty soft class and is denoted by .
Definition 11. Let . If, for all , , then is called a universal soft class and is denoted by .
Definition 12. Let . Then, is a soft subclass of , denoted by , if, for all , .
Example 13. Let be a parameter set, let be an initial universe, and let be a decision makers set. If then soft classes can be written as and .
Note that, for all , since , .
Proposition 14. If , and , then(1);(2);(3);(4) and .
Proof. If , , and , then, for all ,(1);(2);(3);(4) and ; then, and .
Definition 15. Let . Then, and are equal soft classes if and only if and . This relation is denoted by .
Definition 16. Let and and let . Then, according to the soft set , degree of subsethood of soft set , denoted by , is defined as follows:
Here, is set of parameters such that .
Definition 17. Let , let , and let and be two subsets of such that and . If , , and , , then is called almost-subclass of soft class and is denoted by .
From now on, decision makers set will denote set of such that .
Definition 18. Let and let . Then, according to the soft class , degree of subclasshood of soft class , denoted by , is defined as follows: Here, for all , and , such that .
Example 19. Let us consider soft class given in Example 13 and soft class given as follows: Here, since , , and , . Then, Thus, and .
Corollary 20. Let . Then(1)if, , then ;(2)if, may be -subclass of soft class ;(3)if, may not be a subclass of soft class .
Definition 21. Let . Then, union of soft classes and , denoted by , is defined by class of soft sets as follows:
Example 22. Let be a parameter set, let be an initial universe, and let be a decision makers set. If then soft classes can be written as and .
Here,
Proposition 23. If , and , then(1);(2);(3);(4);(5);(6).
Proof. Let , and . Then, for all ,(1), since ;(2), since ;(3), since ;(4), since ;(5), since ;(6), since .
Definition 24. Let . Then, intersection of soft classes and , denoted by , is defined by class of soft sets as follows:
Example 25. Let us consider soft classes and given in Example 22. Then,
Proposition 26. If , and , then(1);(2);(3);(4);(5);(6).
Proof. Let , and . Then, for all ,(1), since ;(2), since ;(3), since ;(4), since ;(5), since ;(6), since .
Proposition 27. If , and , then(1);(2).
Proof. The proof can be easily obtained from Definitions 21 and 24.
Definition 28. Let . Then, soft complement of soft class , denoted by , is defined by class of soft sets as follows: Here, for all .
Proposition 29. If , then(1);(2).
Proof. The proof can be easily obtained from Definition 28.
Proposition 30. If , then(1);(2).
Proof. Let . Then, for all ,(1), since ;(2), since .
Proposition 31. If , then(1), for all ;(2), for all .
Proof. Since are soft sets for all , the proof is clear.
Proposition 32. Let . Then,(1);(2).
Proof. The proof is obvious from Propositions 30 and 31.
Definition 33. Let and let . Then, is called soft partition of soft set if and only if all of the following conditions hold:(1).(2), for all .(3)If and , then , for all .
Definition 34. Let and let . If, for all , then soft class is called soft cover of soft set .
Example 35. Let us consider soft class given in Example 22. Then, soft class is soft cover of soft set given as follows:
Definition 36. Let . If, for all and , , then soft class is called full soft class and is denoted by .
Proposition 37. Let be two soft covers of soft set . Then, is a soft cover of soft set .
Proof. Assume that and be two soft covers of soft set ; then, for all , and . Hence, for all . So, soft class is a soft cover of soft set .
Proposition 38. Let be two soft covers of soft set . Then, is a soft cover of soft set .
Proof. Assume that and be two soft covers of soft set ; then, for all , and . Hence, for all . So soft class is a soft cover of soft set .
Corollary 39. Let . Then, for all , is a soft cover of soft set .
4. Soft Rough Classes
In this section, we define soft rough class and investigate its some properties.
Definition 40. Let . For , parameterized class (-class) of soft class , denoted by , is defined as follows:
Example 41. Let be a parameter set, let be an initial universe, and let be a set of decision makers. Let us consider soft sets , and given as follows: then all of parameterized classes of are as follows:
Definition 42. Let . Then, for and , -lower approximation, denoted by , is defined as follows: -upper approximation, denoted by , is defined as follows: Moreover, the setsare called the -positive region, the -negative region, and -boundary region of . Here is complement of set . If , is said to be -definable; otherwise is called -rough set.
Proposition 43. Let . Then we have for all .
Definition 44. Let and let . Then, soft -lower approximation, denoted by , is defined as follows: Also, soft -upper approximation, denoted by , is defined as follows: Moreover, the setsare called the soft -positive region, the soft -negative region, and soft -boundary region of , respectively. If , is said to be soft -definable; otherwise is called a soft -rough class.
Example 45. Let us consider soft class given in Example 41. Let , . Then,
Lemma 46. Let and let Then, for all and for all ,(1)if , , and ;(2)if , ;(3);(4);(5);(6);(7) .
Proof. Let and let The proofs of (), (), and () are clear from definitions of -upper and -lower approximations: (4)Let . Then, for some . Thus or . So and . To prove the reverse inclusion, assume that . Then, or . So, for some , or and . Thus, . Then we have .(5)Let . Then we have that or . By definition, there exists some such that and or and . So and . Thus, . We concluded that .(6)Let . By definition, there exists some such that and . So . Hence, . Thus, we conclude that .(7)Let . Then we have that and . By definition, there exists some such that and and and . So and . Thus, . We concluded that . To prove the reverse inclusion, assume that ; then such that and . Hence, and . Since and , and in a similar way . We get that . Then, .
Theorem 47. Let and . Then,(1);(2);(3);(4);(5);(6);(7)if , , and ;(8)if , , and .
Proof. By using Lemma 46, the proof can be easily made.
Theorem 48. Let and let . Then,(1);(2);(3);(4);(5);(6);(7) ;(8) .
Proof. (1) It is straightforward.
(2) It is straightforward.
(3) Let . Then, for some , . Since , , and . Therefore and ; From definition of soft -lower approximation
(4) Let . Then, for all , and , for some . Therefore . From definition of soft -upper approximation .
(5) Since and , from , and , respectively. Therefore,
(6) This is similar to proof .
(7) Since , from , . Similarly, . Therefore, .
(8) Let . By definition of soft -upper approximation, there exist some such that and . Hence, we get that either or . Then, or . This shows thatTo prove the reverse inclusion, note that and ; then from () and , respectively. Thus, From (32) and (33),
Definition 49. Let and let . We define These binary relations are called the lower soft class rough equal relation and the upper soft class rough equal relation, respectively.
Theorem 50. Let and let , and . Then,(1);(2), ;(3);(4);(5).
Proof. (1) Assume that ; then . From Theorem 48, we know that . Thus, and so . Conversely, suppose that . From transitivity of , .
(2) Suppose that and ; then, from definition, and . From Theorem 48, and . Hence, we get and so
(3) Let . Then, from definition, . By Theorem 48, and . It follows that . Therefore,
(4) Let and let . From Theorem 48, we get . Thus, and so .
(5) Suppose that and . By Theorem 48, we have Also, since , . Hence, , and so .
Definition 51. Let and let . We define These binary relations are called the lower soft class rough -equal relation and the upper soft class rough -equal relation, respectively.
Theorem 52. Let , and and let . Then,(1);(2), ;(3);(4);(5).
Proof. (1) Assume that ; then . From Theorem 47, we know that . Thus, and so . Conversely, suppose that . From transitivity of , .
(2) Suppose that and ; then, from definition, and . From Theorem 47, So and .
(3) The proof can be made by similar way to proof of () and ().
(4) Let and . Then, and since , . Therefore, . We have .
(5) The proof can be made by similar way to ().
5. Decision Making Using Soft Rough Class
In this section, some concepts are defined to construct a decision making method using soft rough class and a decision making algorithm is given. Then, an application of proposed decision making method is made for a real problem.
Definition 53. Let and let be a soft set (reference soft set) over . Then, consistency degree of soft set related to parameter and soft class , denoted by , is formulated as follows:According to soft class consistency degree of soft set , denoted by , is formulated as follows:
Definition 54. Let and let be a soft set over . Then, relative consistence degree (rcd) between soft class and soft set related to parameter , denoted by , is formulated as follows:Between soft class and soft set total relative consistency degree is formulated as follows:
Definition 55. Let and let be a soft set over . Then is called effectiveness of decision maker and is denoted by .
Now we will give relations between two decision makers in decision maker set .
Definition 56. Let and let be a soft set over . Effectiveness relations between and are defined as follows:(1)If , is more effective than .(2)If , has same effect as .(3)If , is more effective than .
Algorithm 57.
Step 1. Construct a soft class and reference soft set over .
Step 2. Find the consistency degree of soft set denoted by related to parameter .
Step 3. Find consistency degree of soft set according to soft class .
Step 4. Find total relative consistency degree between soft class and soft set .
Step 5. Find effectiveness of each decision maker .
Step 6. Chose effective decision maker.
6. Applied Example
Assume that an investment company wants to employ stock market analysts. Five persons apply for this position in the company and the department of human resources wants to make appropriate choice among the applicants. Therefore, department of human resources wants some previous evaluations made by applicants , and for firms in different times in the last two years.
Step 1. According to the appreciation criteria, evaluations of applicants , , , , and performed in different times specified by human resources department are represented by soft sets , , , , and given as follows:Department of human resources has real results previously obtained in specified times: , , , , and . These real results are represented by soft set (reference soft set) as follows: Tabular representation of soft class is shown in Table 3.
Step 2. Using (37), consistency degree of soft set for time parameters , , , , and is obtained as in Table 4.
Step 3. Using (38) and Table 4, consistency degree of soft set related to soft class is obtained as .
Step 4. Using (39), relative consistency degrees of soft set with respect to soft class are as in Table 5. And, from (40), total relative consistency degrees of soft set with respect to soft class are as in Table 6.
Step 5. Using Definition 55, effectiveness of the applicants , , , , and is obtained as follows:
Step 6. From Definition 56, effectiveness of applicants can be ordered as follows: Then, is the most effective decision maker in soft class by soft set .
7. Conclusion
In this paper, we have defined concepts of soft class, soft class operations, and soft rough class. Then we have presented a decision making method based on the soft rough class. Finally, we have provided an example that demonstrated that this decision making method can successfully work. It can be applied to problems of many fields that contain uncertainty. Next, we can define fuzzy soft class and fuzzy soft rough class and their operations as generalization of soft classes and soft rough classes. Also a reduction method can be developed based on soft class and soft rough classes.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.